Cerenkov radiation and its polarization

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Cerenkov radiation and its polarization

S S LAHOTI and R G TAKWALE

Department of Physics, University of Poona, Poona 411 007 MS received 22 November 1976; revised 18 March 1977

Abstract. General expression for the energy loss in Cerenkov radiation due to a charged particle possessing anomalous magnetic moment is obtained. The expressions include the spin-polarization of the particles. The contribution to the radiation due to anomalous magnetic moment is found to be small as compared to that due to charge.

The interference term of charge and anomalous magnetic moment gives better contribution as compared to the term containing only anomalous magnetic moment. Polari- zation of the radiation as dependent on polarization of beam of particles is studied.

The radiation has a dominant nature of linear polarization with small quantum correct- ions. The spin-flip also gives quantum correction to strong linear polarization and at threshold when phase velocity equals velocity of the particle.

Keywords. Cerenkov radiation; intensity; anomalous magnetic moment; spin-polarization; spin-flip.

1. Introduction

Cerenkov radiation given out by a charged particle moving through a medium with a velocity greater than the phase velocity of light in the medium has been given exten- sive theoretical as well as experimental study by many workers for quite a long time.

After discovery by Cerenkov (1934), Frank and Tamm (1937) gave the classical theory and Ginzburg (1940) and Sokolov (1940) developed the quantum theory of Cerenkov radiation. Bolotovskii (1961) has reviewed the work on this topic. The theoretieai studies are mostly classical and the interest lies in obtaining energy loss due to the radiation in transparent isotropic as well as tmiaxial dielectric media. Quantum electrodynamical treatment for the radiation is developed by considering Dirac particle with quantised electromagnetic field. Quantum mechanical correction to the Cerenkov angle and also the energy loss have been obtained. Nemtan (1953) has obtained formulae for Cerenkov radiation as a first order transition induced by the interaction of charged particles with electromagnetic field of the radiation and the medium. In the first order transition the interaction of the charged particle with the radiation field is considered and that with the atoms of the medium is neglected since it gives rise to radiative transitions in third or higher orders. Further the interaction induces transitions between eigen states which are chosen to be the momentum states of the incident particle (Nemtan 1953).

The quantum electrodynamical treatment of Cerenkov radiation has been developed further by Sokolov and Loskutov (1957), Loskutov and Kukanov (I958) to consider 163

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polarization of Cerenkov radiation due to spin polarization of a beam of electrons.

It is shown that the radiation consists of two parts; a polarized part which vanishes at threshold (i.e. when velocity of particles equals the phase velocity of light in the medium) and unpolarized part which does not vanish at the threshold. The unpolarized part of the radiation is accompanied by the spirt-flip (ss . . . . 1) of the particle.

Kukanov (1961) has considered the motion of an uncharged Dirac particle with a normal magnetic moment and has obtained expressions for the energy loss by a polarized beam of particles and has also considered polarization of radiation. When a motion of the charged particle with a magnetic moment is considered by treating the particle by Dirac equation with the addition of Pauli term for anomalous magnetic moment, it is expected that the effect due to small value of anomalous magnetic moment will obviously be small. The major contribution to the radiation is due to the charge and also the normal magnetic moment of the particle. The product term of normal and anomalous magnetic moment will contribute more than that of the Pauli term.

We have considered the emission of Cerenkov radiation due to a charged particle possessing total magnetic moment (i.e. normal and anomalous) and the general expression for the energy loss has been obtained. The general expression includes all the descriptions of radiation emitted by polarized beam of particle. The polarization of radiation has been studied in some interesting cases.

2. General expression

For the process of Cerenkov radiation due to passage of charged particles with an anomalous magnetic moment through a medium with a velocity greater than the phase velocity of light in the medium, we consider interaction of charged particle with an electromagnetic field of the radiation and the medium with emission of photon.

The conservation laws for the process are:

E = E ' + ~w

p = p' + x (1)

where I x J = ~w/c'; c'---c/n and n is the refractive index of the medium, p and p', are momentum vectors and E = (p2c~-bm~c4)½ and E ' : (p'Sc~-krn2c4)½ are energies of the particle before and after the emission, x --Jl k is the momentum and ~w is the energy of the emitted photon.

The interaction Hamiltonian for the process is the current term with the multiplica- tion of electromagnetic vector potential and can be written as (Muirhead 1968) ( i 1 H ' I f ) = i ~ ( p ' ) [F 1 y~ + i F 2 ¢a~ x ] ~b (p) Art (2) where x~ is the photon four-vector = [x,/iw], a t t = i

[y~y--7~ytt]/2,

Ft= e, the charge on the particle; F 2 ~/za, the anomalous magnetic moment in the units of e~/2mc (m----mass of the paxtiele). Here the total magnetic moment of the particle is

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/ ~ : l + / z ° which in our notation is i~:[Fffe+F2]. For the electron/~ : ( 1 + a/2~r)lz B where a is the fine structure constant and /z B is the Bohr magneton. For the proton t~ = 2.79/~A" where/~Ar : e~/2mc nuclear magneton. Here m =mass of nucleon and F i f e : 1/~N; F2:1.79/z~.. For neutron

/z =--1.91 /~Ar i.e. F i f e : O ; Fz=--1.91/z N.

The second term in eq. (2) is of the order ofF2 xand F2~ 10-SF1 for electron but it is comparable with F 1 for proton. ~(p') and ~b(p) are Dirac wave functions, ~,l* ~ ( r , )4) are Dirac matrices. Al* is the electromagnetic four-vector potential. Here

/ m e ~ ~ ' , r

~(P" : / V E ~ Z a(p') expi (P'-~- ~ t )

(3a) p,

A / m c ~ r

P

Al* = AV/~C'z _{2 w V ~} el* expi ( ~ - - - w t ) (3b) r

k

and p =- P' c and p' = ~p,--j-) are the particle four-mementa before ~ d after emission of photon, el* = (e, e4) is a unit polarization four-vector of photon. Using the relation (Muirhead 1968).

el, [F 1 ~,l* + iF~ ~l*. X.] = [F1 + 2me F,] el* ~,l* + iF 2 el, rl* (4) where

rl* : pl* + pl*. I

Substituting the Dirae wave function ~b and eleetromaguetic potential A~ in eq. (2) and multiplying by complex conjugate we get

m~ c4/~c '~ [ ^{, + ) -} ¢+)1

I(flH'li)l~-

-- 2 E E ' w V Tr t O A O Ap,

J

8p~,, (5) where spin projection operators are:

p \ 2me l

and

,+, = ( m c - - i ~ . p ' ~ ( l + i ~ 6 y a , ' )

Ap, \ 2me /

O : [F 1 + 2m c Fg ]e l* y l* + i F~e ~ r I,

(6b)

0 =740+~'4

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~o and ¢0' are spin four-vectors of the particle before and after radiation respectively.

In the rest system of the particle oJ is a unit vector with vanishing fourth component (Sakurai 1967).

[OJ]rest frame ~--- [~, 0], [O/]rest frame = [~', 0)

where ~ and ~' are unit spin vectors of the charged particle before and after emission respectively.

For any frame of reference, general expression for ~o is given by

[O']g~n,ral =(~, o~)- [~ +

P (P" g') i l

P I s]

₍₇₎

rn(E+mc~) ' mc J"

Carring out the covariant trace calculations we get the general expression for I < f l n ' l i > [ ~

as

-- m'c4h c'Z ~ F12 {2(ep)[(ep)(1*

+(oJ~o'))+(e*oJ)(pto')]

[ ( f l H ' l i ) ] 2 2EE, V w ( 4m2c ,

+

+ (px)[2(eco)(e~o')--(ee)(1

+ (oJo'))]

+ (xo)[(ee)(pco')-- 2(ep)(eo/)]

F1Fg { (ee)(px)(1 +(~oco'))+ 2(px)(e~o)(e~o')--(ee)(po/)(xco)*

m o

+(ep)(xoJ)(eo')-- (ep)(e* oJ)(po/) )* m~Sci ( ep)(ep)(px)(1 + (~o/) )-- 2mic~(px)( e~o)( e* ~o ')* {

--(P~°')(x°J)(m~c~ + (ep)(eP) ) + m2c~(px)( l + (~°~°'))} t"*

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By using energy-momentum conservation the angle 0 of the Cerenkov cone i.e.

angle between p and x is given by (Kunkanov~1961).

c o s 0 = flnL-2ppc ~ 1 -- n--- ~ , (9)

where the second term represents the quantum correction to the usual formula.

eq. (9) and afterwards we use p --- l P I and x ---- [ x I for simplicity.

By the perturbation theory, transition probability per unit time is:

In

fir= 2,tr p(k)

I(fl H ' [ i ) t S.

By considering photon emitted into solid angle d ~ ~-- sin 0

dO

d$ = 2~r sin O

dO

wherein we used azimuthal symmetry, the energy density of the final state is:

- - t/'k 2 ( d E f ~ - l d ~.~ _ g e t

P(~) (2~)~ \ dk / 2~ ~ pc~

- - kdk.

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Therefore, energy radiated per unit time is given by

f - f w e'w w l2.

w = ~ w w = - - - k d k [ < f I

li)

,r pc*- Using k - - w

C'

= f V e ' w z dw [ ( f l U , ] i ) ] z. (10)

W

rr p c s ~ C'*-

If F*- = 0 i.e. if we consider charged particle without anomalous magnetic moment and retain the higher order terms of the type p~ and neglect terms like px, x*- we get.

W r n a x

W - - F 1 2 (e. p) (e. p) [ 1 + w . to + oJ4 w4 ] wdw. (11)

~Ep

0

Averaging over the initial and summing over the final spin states we get:

W _

w l n a x

4~ Ep f (e . p) (e . p) wdw

0

W m a x

_ e% f sin* 0 wdw, 4rr c ~

o

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which is obtained by Nemtan (1953).

Energy radiated per unit time per unit length

W m a x

W' - - e2 f sin*- 0 w dw, 4,n.c ~

0

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which is a Frank and Tamm's (1937) relation when/f -~ 0.

3. Photon polarization

We shall now consider two different cases of polarization namely (i) particle longitudinally polarized (ii) particle transverely polarized and find the polarization effects in expression (10). F o r convenience we shall consider separately the contribution of each term containing F1 ~, FxF~ and F*- ~ in the general expression for W. For this purpose we split Winto three parts namely, WFI', WFtF, and WF,. and write eq. (10) in the form.

W m a x

1 f [wv: + wp~v, + wpo,l, w dw. 04)

W - "- ~'e%r~'-

0

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In order to describe photon polarization we shall assume that the photon is moving along Z-direction (Z ]l x) and particle momentum p is in YZ-plane. Obviously e will be in the XY-plane and

et~ ~ (eos~, sina e in, 0, 0) (15)

where a is the angle between three-vector e arm X-axis and 3 is the phase angle.

The orientation of transverse vector e or the values o f a and 8 decide the nature of polarization of the radiation.

Case (i) Particle longitudinally polarized

For this case, four-spin vectors of the particle in the state before and after radiation

a r e :

_[s'E'~,,ilp'is']

r sE

^ ilplsl;./

mc J oJ' - L-~-~c 2 ~ J (16)

where s, s'=-4-1 and positive sign indicates that particle spirt is in the direction of motion and negative sign in the opposite direction of motion.

On substituting the expression for four-spin vector oJ and photon polarization vector e~ in the expression (14) we obtain

~ x ' ( 1 1 ) [ ( l + s s , ) _ _ _ WF1F2 =F1F2mc ( ~ ~ -- -~]

+pS sin~O sin 2 a)}

W F , = F e S m S c 2 { ~ ( 1 - - 1 ) [ ( l +

2ss'E 2 sin~O sin~a]--ss'x2c°s~O(m2c~

m2c 4 E~

(:Vb)

P=

sin~O sin~a (1 + s s ' ) - - 2ss'E~

) ~ sin'Osina=|

a ",

m~c ~ m c .1 )

(17c)

In the above expressions we have retained terms proportional to p2, p x and x 2 and neglected the terms of higher order of smallness. The first term in (17a) is a dominant one since it is proportional to pZ and the terms in the brackets of (17b) and (17c) are of the order of xL

I f s s ' = - - 1 i.e. when there is a spin-flip, the terms containing (1 +ss') in (17) cancel out but the rest of the terms give small contribution of the order of x 2 and is purely of quantum nature.

For a=~r[2, the expression (17) gives the usual dominant linear polarization per- pendieular to the surface of the cone i.e. along Y-axis (W~), but if a = 0 the expressions give linear polarization along X-axis (W3), i.e. parallel to the surface of the cone and the contribution is quite small. Further a~,r[4 gives contribution due to circular polarization of the radiation. Since phase angle 3 from photon polarization vector does not enter in the above expression we have equal contribution from left (W_t) and right (W+x), circular polarization. Hence W _ I : W+ z. The equal contribution of left and right circular polarization shows th at the radiation is linearly polarized

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and the total intensity of radiation is W : W2,-k W a as given by Sokolov and Loskutov (1957) for the special case of F~--0.

For electron F~ < F 1 and the contribution due to terms containing F2 ~ and

F1F 2

will be very small. However, the contribution due to the interference t e r m

WFzFz

will be larger than that of

WF,"

Case (ii) Particle transversely polarized

For this case the four-spin vectors of the particle before and after radiation are:

o~ : [ s , 0] (18)

o ~ = [ ~ , 0 ] ; ' ^ '

where ~ is a urtit spirt vector perpendicular to the direction of motion of the particle.

For convenience let us take ~ and ~' along X-axis then

ss"

= - k 1 will mean s---k 1 ----s' or s - - - - 1 = s ' i.e.~ there is no spin-flip. The spin vector of the particle is perpendicular to the plane of p and x and is along or opposite to )(-axis.

In this case we get the following expressions:

WFI' FI~ l = ~

2p ~sin g 0sin S

~(l+ss')+-2- x2 (

1 - - - ~ 1)

(2ss'cos2a--(l+ss')) 1

(19a)

I~( 1) (l+ss')-- 2ss' c°s~)l

^(19b)

Pz sin~.Osin~a (l +ss,)] 1 WF~, := F22 m~c2 iX--2 (1--12) I((l + ss')--2ss' cos2a) + m~c2

(19c) In this case of radiation by transversly polarized beam, the Cerenkov radiation is strongly linearly polarized as indicated by first term in (19a). The spin-flip

(ss':--

1) contributes to the radiation and is of the order o f x 2. This is purely a quantum effect.

If a : 0 and

ss':--1

the contribution of linear photon polarization will be only along X-axis. Similarly if a : 0 and

ss'

= - k 1 ; W-- 0 and hence there is no radiation with linear photon polarization along X-axis. A case of no radiation exists when

a:~r/2

and

ss'

: - - 1. But if a=~r/2 and

ss':

+ 1 the radiation will be only along Y-axis i.e. totally linearly polarized. In this case also the sum of linear polarizations along X-axis, W z, and along Y-axis, W~, will give total intensity W : Wg. + W a. I f

a:rr/4

the equal contribution of left, W-l, and right, W+x circular polarization gives total intensity W = W_I+ W+I.

From expressions (19) it follows that the C~renkov radiation vanishes at threshold i.e. at cos0----1 and for a = 0 .

W(ss'=+

1) : 0 and

W(ss'..-:--I)

Wmax

- - X~ - -

--4 ÷ F1 Fz mc q-Fz z m~c~/

21r Ep

0

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which is again a purely q u a n t u m effect i.e. r a d i a t i o n is partially polarized a n d n o n - vanishing at threshold. This result o b t a i n e d f o r general case matches with the result o f L o s k u t o v and K u k a n o v (1958) for their special case (F,z:-0).

T h e expressions and the discussion given a b o v e is applicable to the radiation given by all the c h a r g e d particles with spin half. I n the case o f stable particles like electrons a n d p r o t o n s t h e contributions due to the t e r m s containing F~ 2 a n d F1F 2 in eqs. (17) a n d (19) d e p e n d o n the relative value o f a n o m a l o u s magnetic m o m e n t o f the particle.

F o r electron Fg. -- 10 -3 F1 a n d the c o n t r i b u t i o n for

WF1F2

is larger t h a n WFa 9". H o w - ever, for p r o t o n , F1,~F~ a n d the t e r m

WF1F2

gives contribution to the radiation c o m p a r a b l e to t h a t of WF2 z. T h e n e u t r o n gives out C e r e n k o v radiation due to its a n o m a l o u s magnetic m o m e n t and the r a d i a t i o n is given b y the terms containing F~ ~.

References

Bolotovskii 1961 Usp. Fiz. Nauk. 78 295; 1962 Engl. Trans. Soy. Phys. Usp. 4 781 Cerenkov P A 1934 Compt. Rend. 2 451

Frank and Tamm 1937 Dokl. Akad. Nauk SSSR 14 109 Ginzburg V L 1940 Zh. Eksp. Teor. Fiz. 10 589 Kukanov A V 1961 Opt. Spectrosc. USSR 10 147

Loskutov Yu M and Kukanov A V 1958 Sov. Phys. JETP 34 2 328

Muirhead 1968 The Physics of Elementary Particles (Oxford: Pergamon Press) pp. 280, 512 Nemtan S N 1953 Phys. Rev. 92 1362

Sakurai J J 1967 Advanced Quantum Mechanics (London: Addison-Wesley Pub. Co.) p. 314 Sokolov A A and Loskutov Yu M 1957 Zh. Eksp. Teor. Fiz. 32 630

Sokolov A A 1940 DokL Akad. Nauk. SSSR 28 415

Cerenkov radiation and its polarization